UC-NRLF 


tDO 


8S* 

££   5 


II 


_ 
O  < 

z  tr 


WORKS   BY 

L.  A.  WATERBURY 

PUBLISHED    BY 

JOHN    WILEY    &     SONS 


Cement  Laboratory  Manual. 

A  Manual  of  Instructions  for 
the  Use  of  Students  in  Cement 
Laboratory  Practice.  12mo, 
vii  +122  pages,  28  figures. 
Cloth,  $1.00. 

A     Vest  =  pocket     Handbook     of 
Mathematics  for  Engineers. 

vi  +  91  pages,  61  figures.  Mo- 
rocco, $1.00  net.  '' 


A  VEST-POCKET 

0 

_j 
< 

HANDBOOK 

O  C 

<T    5 

OF  O 

MATHEMATICS 

§s 

FOR  5° 

ENGINEERS 


BY 
L.  A.  WATERBURY,  C.E. 

U 

Professor  of  Civil  Engineering,  University  of 


5'Sv 

OF  THE  \ 

UNIVERSITY  ) 

,  or 

FIKST    EDITION 


FIRST    THOUSAND  5   - 

ji 


NEW  YORK 

JOHN  WILEY  AND  SONS 

LONDON:  CHAPMAN  &  HALL,  LIMITED 

1908 


GENERAL 


COPYRIGHT,  1908, 
BY 

L.  A.  WATERBURY 


Stanhope  Ipress 

F.    H.   GILSON     COMPAN1 
BOSTON.     U.S.A. 


PREFACE. 


Tliis  handbook  is  intended  as  a  reference  2  oc 

book,  for  the  use  of  those  who  have  studied  >  m 

or  are  studying  the  branches  of  mathematics  <  § 

usually  taught  in  engineering  courses.     It  is  <  £} 

not  intended  for  a  text  book,  and  does  not,  ^ 

therefore,   attempt    to    prove  many  of    the  _j 

formulae  which  are  given.  <  m 

Most    of    the  material  in  this  book  was  z  3 

obtained  from  the  following  sources:  algebra  S  Q 

from  Hall  &  Knight's  Algebra  (Macmillan  {Jl  ^ 

Co.)  ;     trigonometry    from     Bowser's    Trig-  -jj  ° 
onometry;   analytic  geometry  from  Candy's 

Analytic  Geometry ;  calculus  from  Taylor's  ; 

Differential  and  Integral  Calculus ;  theoret-  •    <  '£ 

ical  mechanics  from  Church's  Mechanics  of  o  D 

Engineering ;    and    mechanics    of   materials  £  H 

from  Merriman's  Mechanics  of    Materials  ;  -  o 

to  all  of  which  the  writer  is  very  much  in-  — s= 

debted   and  from  all  these  Authors  he  has  '      <  a 

received  permission  to  use  the  material.   The  -  £ 

reader  is  referred  to  these  works  for  the  proof  ;      £  ^_ 

and  explanation  of  the  various  formulae.  -  t; 

L.  A.  W.  _  h  2 
TUCSON,  ARIZ.,  March,  1908. 


179775 


CONTENTS.  ,  > 

PAGE  §  j- 

ALGEBRA  .    .  1         £  2 

H  O 

TRIGONOMETRY 5 

Plane  Triangles 8 

Spherical  Triangles 9        ^£ 

>    UJ 

ANALYTIC  GEOMETRY 11        ^5 

Transformation  of  Coordinates.    ...  11         %  \n 

The  Straight  Line 13  ~ 

The  Circle 14 

The  Parabola 15          < 

The  Ellipse 15          |  § 

The  Hyperbola 16          :    5 

The  Cycloid 17          =   j 

Miscellaneous  Curves 17          ~   '3 

Solids 18 

DIFFERENTIAL  CALCULUS 20         j  w 

INTEGRAL  CALCULUS 23         §  ^ 

UJ    Q 

THEORETICAL  MECHANICS 36         b  J 

-  o 

Statics:  --— 

Equilibrium  of  Forces 38  ^  tf 

Center  of  Pressure 39  -  ^ 

Center  of  Gravity 40  :  < 

Moment  of  Inertia 42  5  { 

Product  of  Inertia 44  I  ^ 

Radius  of  Gyration 45 

Ellipsoid  of  Inertia 45         — * 

Dynamics:  w  ! 

Velocity  and  Acceleration      ....  46  • 

Falling  Bodies 46  :    - 

Impact 46 

Virtual  Velocities 47  5  [ 

Curvilinear  Motion 47 

Projectiles 48 

v 


VI  CONTENTS 

PAGE 

Translation 49 

Rotation 49 

Center  of  Oscillation 50 

Pendulum 51 

Work,  Energy,  Power 51 

Friction 51 

MECHANICS  OF  MATERIALS 53 

Direct  Stress 54 

Eccentric  Loads 55 

Equation  of  Neutral  Axis 58 

Kernel  or  Core-Section 58 

Section  Modulus  Polygons 59 

Diagonal  Stresses 61 

Pipes,  Cylinders,  Spheres 62 

Riveted  Joints 63 

Beams: 

Vertical  Shear 64 

Shearing  Stresses 65 

Bending  Moment 65 

Theorem  of  Three  Moments  ....  66 

Flexural  Stresses 67 

Table  of  Beams 68 

Struts  and  Columns: 

Euler's  Formula       81 

Rankine's  Formula '.  82 

Ritter's  Formula 83 

The  Straight  Line  Formula    ....  83 

Engineering  News  Formula   ....  84 

Eccentrically  Loaded  Columns  ...  86 

TORSION 86 


ALGEBRA. 


EXPONENTS   AND   LOGARITHMS.  : 

If  aw  =  b,  m  =  loga  6 .     am  .  an  =  am+w,  - 

.'.  log  x  .  y  =  logo:  +logy.     aTOH-an  =  am~n»  <g 
.'.  log  (z-^2/)  =  log:r-log2/. 

-   09 

.'.  log  x2  =  2  .  log  x.      (awl)n  =  aw  •  n,  ^    : 

/.  logxft  =  n  .logx.     a°=l,  :i  '] 

^  < 

/.  log(l)=0.  F° 

The  base  of  the  common  system  of  loga-  j  w 

rithms  is  10.  $  3 

The  base  of  the  natural  system  of  loga-  3  Q 

rithms  is  C  d 


,4-+  .  .  .  =  2.7182818284. 


The  cologarithm  of  a  number  is  the  loga- 
rithm of  its  reciprocal.     Log  f-J  =  0  —  log  x. 

To  transform  a  logarithm  from  base  e  to 
base  10,  multiply  by  logio  e. 

Log10  e  =  0.43429448. 

Log.  10  =  2.30258509. 

1 

"  log«  10  ' 
1 


2  ALGEBRA 

QUADRATIC   EQUATIONS. 


2a 


PROPORTION. 

If  a  :  b  :  :  c  :  d, 

a      c  b      d 

7-— j*       or       -  =  -• 
b      a  a      c 

ad  =  bc,          ^=^±-d, 
o  d 

a  —  b _ c—d      a+b _ c+d 
b  d         a—b      c—d 

ARITHMETICAL  PROGRESSION. 

a,  a+d,  a  +2  d,  .  .  . 
Last  term,  L  =  a  +  (n  —  1)  d. 
Sum  of  terms, 


GEOMETRICAL   PROGRESSION. 

o,  ar,  ar2,  ar3,  .  .  . 
Last  term,  L  =  arn~l. 
Geometric  mean,  M  =  ^ab. 


Sum, 


rL-a 


1-r  r-1 

For  an  infinite  geometrical  series,  the  sum 
to  infinity  is  S=    _  •  • 


• 


ALGEBRA 

HARMONIC   PROGRESSION. 

b,  c  are  in  harmonic  progression  if 
a  _  a  —  b 


. 


if  _,      ,  _  are  jn  arithmetical  progression. 
a    b    c 


PERMUTATIONS   AND   COMBINATIONS. 

ab  and  ba  are  two  permutations  but  only 
one  combination. 

The  number  of  permutations  possible  of 
n  things  taken  r  at  a  time  is 

nPr  =  n  (n-1)  (n-2)  .  .  .  (n-r  +  1). 

"P-=ln. 

([n  =1X2X3X4  .  .  .  Xn). 


BINOMIAL   THEOREM. 

+b)n  =  an+n  .  an-1  .  b 


SERIES. 

1.  An  infinite  series  in  which  the  terms 
are  alternately  positive  and  negative  is  con- 
vergent if  each  term  is  numerically  less  than 
the  preceding  term. 


4  ALGEBRA 

2.  An   infinite  series  in   which  all  of  the 
terms  are  of   the  same  sign  is   divergent  if 
each  term  is  greater  than  some  finite  quan- 
tity, however  small. 

3.  An  infinite  series  is  convergent  if  from 
and  after  some  fixed  term  the  ratio  of  each 
term  to  the  preceding  term  is  numerically  less 
than  unity. 

4.  An  infinite  series  in  which  all  the  terms 
are  of  the  same  sign  is  divergent  if  from  and 
after  some  fixed  term  the  ratio  of  each  term 
to  the  preceding  term  is  greater  than  unity, 
or  is  equal  to  unity. 

5.  If  there  are  two  infinite  series  in  each 
of  which  all  of  the  terms  are  positive,  and  if 
the  ratio  of  the  corresponding  terms  in  the 
two  series  is  always  finite,  the  two  series  are 
both  convergent,  or  both  divergent. 


DETERMINANTS. 

|at     &i|=a6  _Q  b 

'O2       b2\ 


ai  61  ci 

tt2    b2    C2 

as  63  03 


~d\  •  &2  •  C 3  + 
«2  •  b3  •  Ci  + 
03  .  61  .  C2 


—  CL2  •  bi  .  C3  —  03  .  &2  •  Ci  . 


then 


a^x  +  b%y  +  ciz  4-  dz  =  0, 
a^x  +  b$y  +  c&z  +  ds  =  0, 


3;  =        -y        = 


-1 


TRIGONOMETRY. 


r*  C 

2 

Radius  =  1  .                                                                    > 

E                Q             F      2  PC 

AB  —  sin  0. 

;>\ 

/         h-  r- 
./            >   UJ 

OA=cos  0. 

^ 

xp         <| 

\                        '     J 

CD  =  tan  0. 
EF  =  cotO.                 o 

X 

N 

AC                   J 

OD  =  sec0.                             F'g- 
OF  =  cosec  0. 

i! 

s\ 

BG  =  covers  0=1— sin  0. 

.      sin  0 
tan  0=-— H' 


sin2 

sec2  0  =  i 
cosec2  0  =  l+cot20. 
exsec  0  =  sec  0-1. 


For  0  in  radians, 


05  07 


02  04  06 

~+~ 


03          2.05  1707 

tan  0  =  0  +  T  +3^5  +373-75^7 


TRIGONOMETRY 


Sli 


TRIGONOMETRY  7 

sin  (A  +B)  =sin  A  .  cos  B  +cos  A  .  sin  B. 
sin  (A—B)=  sin  A  .  cos  B  —  cos  A  .  sin  B. 
cos  (A  +B)=  cos  A  .  cos  B  —  sin  A  .  sin',B. 
cos  (A  —  J5)  =cos  A  .  cos  5  +sin  A  .  sin  B. 
tan  A  +  tan  J? 

1_tanA.tana-  oS 

tan  A  -  tan  B  r  ~ 

(A-B)^1+tanA-tanB-  ;;  j 

z  ^ 

sin  2  A  =2  .  sin  A  .  cos  A.  <  5 

cos  2  A  —  cos2  -4  —  sin2  A 

=  2  cos2  ,4-1  <„, 


2  .  tan  A 

tan  2  A  —  -  —  -  —  z—A  • 
I  —  tan2  A 


/A\  _  1  —  cos  A 

sin  3  A  =3  .  sin  A  — 4  .  sin3  * 
cos  3  A  =  4  cos3  A  —  3  cos  A . 
3  tan  A  —  tan3  A 


tan  3  A  = 


1-3  tan2  A 


sin  A  +sin  B  =  2  .  sin  —  -  —   .cos  —  ^  —  • 


sin  A  —sin  B  =  2  cos  —  ~  —   .sin  —  pr—  • 


A+B          A-B 
cos  A  +  cos  B  =  2  cos  —  s  —  .cos  —  jr  -- 


.A+B      .    A-B 
cos  A  —  cos  B  =  —  2  sm  —  ~  —  .  sin  —  r  —  • 


TRIGONOMETRY. 


sin  A  +sin  B  =  tan  £  (A  +B) 
sin  A  —sin  B      tan  £  (A  —  B) 


sin  A  +  sin  B 
cos  A  +  cos  B 


=  tan  £  (A+B). 


sinA+sinB 
cos  A  —  cos  B 

sin  A  —sin  B  _         ,   ,         „. 
cos  A  +cos  B  ~~ 


sin  A  —sin  B 
cos  A  —  cos  B 


=  cot  £ 


cos  A  4- cos  B 
cos  A  —  cos  B 


rv- 


PLANE  TRIANGLES. 


sin  A  +sin  B 
+sin  C 

=  4  cos  —  .  cos 


Fig.  2. 


cos  A  +cos  B  +cos  C 


=  l+4.sin| 


.    C 


tan  A  +tan  B  +tan 


tan  A  .  tan  B  .  tan  C. 


a 

sin 


A      sin  B      sin  C 

a2  =  &2+c2_2  .  6  .C  .  COS  , 


= 
a-6      tan  \  (A-B)' 


TRIGONOMETRY 

Area  =  i  b  .  c  .  sin  A 
_  a?  sin  B  .  sin  C 
2  .  sin  A 


=  \/s(s-a)  (s-6)  (s-c), 
where  s  =  i(a+6+c). 

SPHERICAL  TRIANGLES.  '-  £ 

>    UJ 

Center  of  sphere  is  at  0.  ^  S 


?c 
/o 

B 

Fig.  3. 

Right   Spherical    Triangles.     Let    C    repre- 
sent the  right  angle, 

cos  c  =  cos  a  .  cos  6. 
sin  6  =  sin  B  .  sin  c. 
tan  a  —  cos  B  .  tan  c. 
tan  a  =  tan  A  .  sin  b. 

tan  A  .  tan  B  = • 

cos  c 

cos  A  =sin  B  .  cos  a. 

OBLIQUE   SPHERICAL  TRIANGLES. 

sin  a       sin  b       sin  c 

— — 7-  =  — — p>  =  - — 77  =  modulus. 

sin  A       sin  B      sin  C 

cos  a  =  cos  b  .  cos  c  +sin  6  .  sin  c  .  cos  A. 
cos  A  =  —  cos  B  .  cos  C  -f  sin  B  .  sin  C  .  cos  a. 
cot  a  .  sin  6  =  cot  A  .  sin  C  +  cos  C  .  cos  6. 
Let 


10 


TRIGONOMETRY 


then 


s-c) 


sin  6  .  sin  c 


/A\  _  A/sin  s  .  sin  (s  —  a) 
\2  )  ~~  sin  6  .  sin  c 


tan  (-\  =  \/sin  (*-fr)  •  sin  (*  — c)  . 
\2  /  sin  s  .  sin  (s  —  a) 


sin    -    =  V    -  : 


sin  B  .  sin  C 

/cos  (S-B)  .  cos  (S  -  C) 
sin  B  .  sin  C 

/_     cos  S.  cos  (S  -A) 
cos  (S  —  B). cos  (S-Cy 


ANALYTIC   GEOMETRY. 


TRANSFORMATION  OF  COORDINATES. 

To  transform  an  equation  of  a  curve  from 
one  system  of  coordinates  to  another  system, 
substitute  for  each 


variable  its  value  in         ^ 

< 

P 

terms  of   variables 
of  the  new  system. 

^    /p  f  >l 

- 

Rectangular  Sys-    * 
tern.  Old  Axes  Par- 

: 1 

allel  to  New  Axes.       X 

Fig.  4. 


Rectangular  System.     Old   Origin   Coincident 
with  New  Origin. 


Fig.  5. 

xf  =  x  .  cos  9+y  .sind. 
y'=y  .cosQ  —  x  .  sin  6. 
x  =xf  .  cos  0—  yf  .  sin  0. 


y  =!/     cos  O+x*  .sin 
11 


12  ANALYTIC  GEOMETRY 

Rectangular  System.  Old  Axes  not  Parallel 
to  New  Axes.  Old  Origin  not  Coincident 
with  New  Origin. 


Fig.  6. 


xf  =  (x-K)  cos  0  +  (y-k)  sin  0. 
yf  =  (y-k')  cos  O-(x-h)  sin  9. 
x  =  xf  .  cos  0  —  yf  .sin  0+h. 
V  =yf  .  cos  0  +xf  .  sin  0  +k. 

Polar  and  Rectangular  Systems. 


x  =  p  .  cos  0. 
y  =  P  .  sin  0. 

P  =  vV+2/\ 

tan  9-H. 


Fig.  7. 


sin0  = 


cos  0  •• 


^x*+v*' 


v5T^' 


cot  0  =  -  . 

y 

8ec*  =  ^E±Z2. 
cosec  5  = 


/x*+tf 

y 


ANALYTIC  GEOMETRY  13 


THE   STRAIGHT   LINE. 

Equations  of  Straight  Line.  An  equation  of 
the  first  degree  containing  but  two  variables 
can  always  be  represented  by  a  straight  line. 

The  equation  of  the  straight  line  may  as- 
sume the  following  forms,  for  the  rectangular 
system  of  coordinates. 

Ax+By+C=0    ....     (1) 
y  =  mx+k    ......     (2) 

in  which  m  is  the  value  of  the  tangent  of  the 
angle  which  the  line  makes  with  the  X-axis, 
and  k  is  the  intercept  on  the  Y-axis  between 
the  line  and  the  X-axis. 

y-y'  =  A(x-x')       ...     (3) 

in  which  x'  ',  y'  are  the  coordinates  of  a  point 
of  the  line,  and  A  is  a  constant. 


in  which  xr,  y1  and  x"  ',  y"  are  the  coordinates 
of  two  points  of  the  line. 
The  polar  equation  of 
a  straight  line  is 

p  .  cos  (0-a)=&     (5) 

where  k  is  the  length  of 
the  normal  ON. 

Distance  between  Two  Points.  The  distance 
between  two  points,  x',  y'  and  a/',  y",  is 
equal  to 


The  distance  between  two  points,    pi,   #1, 
and  02,  02,  is  equal  to 

^Pl2  +  P22  ~  2  Pl  .  p2  .  COS  (0i  -  02)  • 

.  : 


14 


ANALYTIC  GEOMETRY 


Angle  between  Two  Lines.  The  angle  be- 
tween two  lines,  y  =  m'x+kf  and  y  =  m"x+W ', 
is  the  difference  between  the  two  angles 
whose  tangents  are  m'  and  m". 

Area  of  Triangle.  The  area  of  the  triangle 
whose  vertices  are  (x\,  2/1),  (x2, 2/2),  and  (x3, 2/3), 
is  equal  to 

i  r^ 

*3  2/3  1 

THE   CIRCLE. 

The  most  general  equation  of  the  circle, 
for  rectangular  coordinates,  is 


in  which  a,  b  are  the  coordinates  of  the  cen- 
ter of  the  circle,  and  R  is  the  radius. 

The  following  are  special  equations  of  the 
circle  for  rectangular  and  polar  systems  of 
coordinates. 


2  R  .cos  B. 


X— 


Fig.  10. 


2R  .sin0. 


ANALYTIC  GEOMETRY 


15 


THE   PARABOLA. 

If  the  Y-axis  coincides  with  the  directrix, 
DM,  then 


Q 

7 

TP 

D 

O 

\ 

1 

V 

If  the  F-axis  coincides  with  ON,  passing 
through  the  vertex,  then 

In  Fig.  12,  F  is  the  focus,  OF=OD=a,  and 
L  —  L  is  the  latus  rectum  =  4  a. 

Eccentricity,  e—  -p^  =1. 

THE   ELLIPSE. 


D         e 


Fig.  13. 

F,  F  are  foci. 

Eccentricity,  e<l. 

The  area  of  the  ellipse  is  equal  to  nab. 


16  ANALYTIC  GEOMETRY 

THE  HYPERBOLA. 


Fig.  14. 

A  -A  =  principal  hyperbola. 
B~B  =  conjugate  hyperbola. 
c—c  —  asymptote. 

Principal  hyperbola: 


Asymptotes:    —  -  ^  =n 
a2      b2 

Conjugate  hyperbola:    -  -  £  = 
a2      b2 

When  referred  to  the  asymptotes  as  axes, 
the  equations  become: 

Principal  hyperbola:   xy  = 


Conjugate  hyperbola:  xy=  -  (a<i+b\ 


D-D  is  the  di- 
rectrix. 

F,  F  are  foci. 

FP 
pQ-Ol. 


Fig.  (5. 


ANALYTIC  GEOMETRY 
THE   CYCLOID. 


17 


Fig.  16. 

x  =  a  (0  —  sin  0), 
y—a  (1  —cos  0), 


x  =  a  ,  vers"1  \\  -  ^2  ay-y*- 
THE   SPIRAL   OF   ARCHIMEDES. 


THE  RECIPROCAL    OR  HYPERBOLIC 
SPIRAL. 

I 


THE   PARABOLIC   SPIRAL. 


THE   LITUUS  OR   TRUMPET. 


THE   LOGARITHMIC    SPIRAL. 

log  p  =  k  .  9. 

If  fc  =  l,  and  logarithms  to  the  base  a  are 
employed,  then  the  equation  may  be  written 


18  ANALYTIC  GEOMETRY 

THE   CATENARY. 


THE   CUBIC   PARABOLA. 

THE   SPHERE. 

For  the  origin  at  the  center, 


where  R  is  the  radius. 


CONES. 


The  equation  of  the  cone  generated  by  the 
hne,  z  =  mx+c,  rotated  about  the  Z-axis,  is 


OBLATE   SPHEROIDS. 

The  equation  of  the  oblate  spheroid  gen- 
erated by  the  ellipse,  g  +  g  =  l,  rotated  about 
its  minor  axis,  is 


PROLATE   SPHEROIDS. 

The  equation  of  the  prolate  spheroid  gen- 
erated by  the  ellipse,  g  +  ~  =  1,  rotated  about 
its  major  axis,  is 


x2       y-       z"i 

^_    i    «_  _i_       i 

A2   "*"   A2    "T"  Z9  —  A  • 


§    ANALYTIC  GEOMETRY  19 

HYPERBOLOIDS. 

The  equation   of    the   hyperboloid   of  one 

x2     z2 
aappe,  generated  by  the  hyperbola,  --^=1, 

rotated  about  its  conjugate  axis,  is 

£2  4.  £  _  *  =  i 
a2      a2      62 

The   equation   of    the  hyperboloid   of   two 

x2     z2 
nappes,  generated  by  the  hyperbola,  —2  -  p  =  1 , 

rotated  about  its  transverse  axis,  is 
a2~P  ~P=1* 

THE  PARABOLOID 

The  equation  of  the  paraboloid  of  revolu- 
tion generated  by  the  parabola,  x2  =  4a« 
rotated  about  its  axis,  is 


GENERAL  EQUATION   OF  CONIC 
SECTION. 

The  general  equation  of  any  conic  section, 
for  which  the  Y-axis  coincides  with  the 
directrix  and  the  X-axis  passes  through  the 
foci  normal  to  the  directrix,  is 


where  k  is  the  distance  from  the  directrix  to 
the  focus,  and  e  is  the  eccentricity. 


DIFFERENTIAL 
CALCULUS. 


Variables  will  be  represented  by  u,  v  x  y 
and  0,  and  constants  by  a,  6,  m,  and  'n.  ' 

D  will  be  used  as  the  sign  for  the  deriva- 
tive, and  d  as  the  sign  for  the  differential. 

Sm   i  x  =  angle  whose  sine  is  x. 


D  <&>- 


dx 


.'.  To  obtain  the  derivative  of  any  func- 
tion, drop  the  differential  of  the  variable 
from  the  differential  of  the  function. 


d  (av)=a  .dv. 
d  (u  +v  +x)  =  du  +dv  +dx. 
d(x  .y)=y  .dx+x.dy. 
d(u.v.x.y.  ..)  =  (v.x. 

(u.x.y  .  .  .-)dv  +  (u.v. 

(u  .  v  .  x  .  .  .)  dy  +  .  .  . 


20 


DIFFERENTIAL  CALCULUS        21 

dx 
dx*  =  y  .  xV~l .  dx  -\-x*  .  Iog0  x  .  -^  » 

where  M  =  logo  e. 

d  (&")=&*. iog«b.^ • 

dxa=a  .  xa~l  .  dx. 


d  (sin  x)  =  cos  x  .dx. 

d  (cos  x)=  -sinx  .  dx. 

d  (tan  x)=sec2x  .  dx. 

d  (cot  x)  =  —  cosec2  x  .dx. 

d  (sec  x)  =sec  x  .  tan  x  .  dx. 
d  (cosec  x)  =  —  cosec  x  .  cot  x  .  dx. 

d  (vers  x)  =  d  (1  -  cos  x)  =  +sin  x  .  dx. 
d  (covers  x)=d  (l-sinx)=  -cosx.dx.         j  « 
dtsin-x^dx/vT^  II 

d(cos-1x)=-dx/Vl-x2.  z  < 

d  (tan-1x)=dx/(l+x2).  — 

d(cot~1x)=-dx/(l+x2).  j  eo 

d(sec-1x)  =  dx/(xV/x2-l).  ;  g 
^  ^ 

d  (vers-1  x)  =  dx/v/2  x-x2.  2  O 

d  (covers  -1  x)  =  -  dx/v^x-x2.  _  H  s 

^     "*i 
To  differentiate  a  junction  :  M  « 

1.  Find  the  value  of  the  increment  of  the 
function   in    terms   of   the  increments  of  its 
variables ; 

2.  Consider  the  increments  to  be  infinitesi- 
mals, and  in  all  sums  drop  the  infinitesimals 
of   higher   order  than   the  first,   and  in  the 


22        DIFFERENTIAL  CALCULUS 

remaining  terms  substitute  differentials  for 
increments. 

For  the  maximum  value  of  a  function  the 
first  derivative  is  zero,  and  the  second  deriv- 
ative is  negative. 

For  the  minimum  value  of  a  function  the 
first  derivative  is  zero,  and  the  second  deriva- 
tive is  positive. 

If  -~  assumes  the  form  -•    then 

Fx      D  (Fx) 
fx  ~  D  (/*    ' 


The  radius  of  curvature  for  a  curve,  y  =  fx,  is 


=       =  ^ 

da.  d?y  dx  .  d?y 

(dx? 

where  s  is  length  of  curve. 


INTEGRAL  CALCULUS. 


I  dx=x+C,  where  C  is  the   constant  of 

integration.     The  constant  C  must  be  added 
to  all  of  the  following  forms. 

J  (dx+dy+dz  .  .  .)  = 

Cdx+Cdy+fdz  +  ... 

Cx».dx=^- 
J  n+1 

dx     . 


x  or  vers  x. 


A*.  <**-*. 

J  loge  a 

\  e*  .  dx  —  e*. 

I  ax  .  loge  a  .  dx  =  a*. 

I  sin  x  .  dx=  —  cos 

I  cos  x  .  <£c  =  sin  x  or  —  covers  x. 


cosec2  x  .  dx  =  —  cot  x. 


I  sec  x  .  tan  x  .  dx  =  &ec  x. 


23 


24  INTEGRAL   CALCULUS 

J  cosec  x  .  cot  x  .  dx  =  —  cosec  x. 
J  tan  x  .  dx  =  log  (sec  x). 

i  cot  a;  .  dx  =  log  (sin  x). 

J  cosec  x  .  dx  =  log  (tan  |)  • 
J"sec  x  .  dx  =  \og  [tan  ^  +  1)]  • 


=4—©- 

/(fa  1      ,      /x-a\ 

^^  =  2^-loSC-+i)' 


=log  (a; 

f  _  rf^  1  ,  /x\ 

\  -  ,  =  -  .  sec  -1  (  -  )  , 

J  x\/x2_a2      a  \a/ 


-x2 

=  —  covers  - 


Cf 


INTEGRAL  CALCULUS 


C,   if 


25 


I  a  .  dx  =  a  I  dx. 

fo-c. 

\  x  .  dy  =  xy  —   I  y  . 


dx. 


i^t!  ~&&+**-i  .  log  (a+te)]. 

/a  .  dc  '       If,       ,     ,  ,    .    ,       a     "I 
__^_2  =  _|_log(a+6x)  +  ___J 


«2    1 

a+bx] 

C       dx  I 

J  x(a+bx)  ~  ~  a  '  log 


a+bx 


1          _!     j      /a+&a:\ 

a(a+6x)       a2  "    3g  V    a;    /' 


_      , 

+'     g 


axa?'         \     a? 


dx  1  .  /  A  /b\ 

—  rr-  o=  —F=  •  tan"1  (x  V  -V 
a+6x2      Vafr  >          a/ 


when  a  >0  and  6  >0. 


26  INTEGRAL  CALCULUS 

dx                 1                    <Sa+x  ^~b 
t  ,  ^.2  ~      / •  l°g  -/= /==- » 

when  a>0  and  6<0. 

dx 


r_dx_  x       _      J_   f_c 

J  (a  +6x2)2      2  a  (a  +6x2)      2  a  J  a  J 


2+6x2 

dx 


(a  +6x2)n+1      2na'  (a  +6z2)n 
,  2n-l       f       dx 


L^l       f_ 

2  no       J  (a 

J  a+6x2  = 


a;      a 
6 


(a+6x2)n+1      2  rib  (a 

^J_        f       dx 

2n6      J  (a+6x2)*' 

/dx  J_         /    x2    \ 

x(a+6x2)      2  a     SVa+6x2/ 

r ^__=_j__^r 

J  x2  (a  +6x2)          ax      a  J  « 


a+6x« 


f_        da;  =  !   C         dx 

J  x2  (a  +6x2)n  +1      a  J  x2  (a  +6x2)* 

_6    C         dx 

aj  (a+6x2)n+1 ' 


6  (nP+m+1) 


INTEGRAL  CALCULUS  27 


nP+m+l 

o 
nP+ 


T^i   (V  .  (a  +bxn)P~i  .  dx, 
+m+lJ 


a  (ro  +  1) 

xm+n. 


b  (nP  -\-  Tfi  -f-  n  -f- 1 )  /* 
o  (w  + 1)          I 

^+1-(« 


on  (P -HI) 

nP+m+n  +  1 
an(P-fl) 


-A.    f- 

2a  J  < 


1562 

/a+bx .  dx  — 


105  63 


28  INTEGRAL  CALCULUS 

* .  dx        2  xn^a+bx 

n~l .  dx 


/g.  rfx    _ 
Va+6x 


2  na 
(2n+l)  6 

2(2a- 


fxn~l . 
J  V£+ 


/a+bx 
when  a  >0, 

2 

or  =     . 

when  a<0. 
dx 


f 


/a+bx 


xnVa+bx          (n-l)axn~ 

(2n-3)  6    f          dx_ 
(2n-2)  a  J  ^n-iVa 


+62; 


INTEGRAL  CALCULUS 


29 


fa;2  *S 


/z2  .  dx  x*/~»  -  *  ,  °2 

,  =  -  ~Va2_a;2+        s 

Va2-^2          2  2 

f__^_  x 

J  (a2-z2)i      a2  Vaz-a;2' 

f(a2-rc2)5.^  = 


(5  a2 


dx  1     ,      (         x          \ 

-  -  --  =  -  .  log  1  -  .  I 

x  Vxz  +az      a          \a  +  V  x*  +  a?' 


S£ 
21 

£  u. 
0 


30  INTEGRAL  CALCULUS 

C       dx         _ 


2a?x2          2  a3 


1    . 
log 


Cx 


•dx=  |(2*2±a; 
a4 


L-<fl  -a  cos"1-- 


;2—  a  •  log  - 


f  - 

:  Jvs 

r  _j 


-  log  (x  +  ^x2  ±  a 


,  4-  log  (x-l-^a^ia2). 

^(2a;2±5a2)  Vx2±a2 


INTEGRAL  CALCULUS  31 

/;* 


,     , -  =  vers~1-- 

V2ax-x2  a 


/xmdx       =      xm~1^/2ax-x2 
\/2ax-x2 


(2m- 


i-l)qj^»-'.<fa 


dx 


m  — 

"(2m 


-l        C 

-1)  a  J  xm-i 


dx 


/ 
/ 

/ 
/. 


ax-x*  .  dx 


,  a2    .      ,  x  —  a 
+  2Sm      —  • 


a;wV2ax-rc2.  rfa:=  -  ' 


(2m+l)  a 
m+2 


Cxm-it 


= 


(2m-3)axm 


(2m 


-3        C^Zcuc  —  a? 
-3)aJ         xw~i      *  d:C' 


v  ax2  +6x  +c 

-7=  log  (2  ax  +6  4-2  Va  v/ax2+6x+c). 


dx 


/v/0^+6^+"c  .  dx=  2a*+b  Vax*+bx+c 
4.a 

_/&2_-4_ac\    /*_ 
V     8a     / J  x 


32  INTEGRAL  CALCULUS 

C  dx  =  _JL_ gin_1  /  2aa;-6   \ 

7.  da;  = 


f 
/ 


4  o 

62+4oc 
8a 

a?dz 


n:    fe 
T2^ 


^   r 


3  a 
c.  da:. 


sin  (2  aO. 


Jsin2a:.da;=|-|sin 

/cos2  a; .  dx=  §  +  T  sin  (2  a;). 
2      4 

I  sin2  a; .  cos2  x  .  dx=  -(x  —  -  sin  4  arV 

C  C         dx 

\  sec  a; .  esc  x  .  dx  =    \  —. — 
J  J  sin  x  .  cos  x 

=  log  tan  x. 

Csec*x.csc*x.dx=    C  .   ,    dx — r- 
J  J  sin2  x  .  cos2  x 


=  tan  x  — cot  x. 


m+n 


H I  sin"1"2  x  »  cosn  x  .  dx, 

m  +nj 


INTEGRAL  CALCULUS  33 

sinw+1a;.  cosn~1x 


i-i  r  . 

—, —   I  si 
i+nj 

f+tmt 

m-l   C  • 

I  smm~ 

m    J 


H • —   I  sinw  x  .  cosw~2  x  .  dx. 

+      ' 

dx  = 


m 

cosn  x  .  dx  - 
sin  x . 


c .  cosn-1  x  ,  n  —  1    C 

H I  cos     l  x  .  dx. 

n  n    J 

/sin**  x  ,    _ 
cosnx 

sinw+1  x  n  —  m  —  2   rsinmx. 

(n  —  l)cosn~1x          n  —  I     J    cos""2 

/cosn  x     , 
.   m    .  dx  = 
sin    x 

-n  —  2   f*cosnx 
i  —  \     J    sinm  ~ 


dx 


(m  —  1)  sin"1"1  x          m- 


/dx  —  cos  x  m  —  2  C      dx  

sinm  x      (m  —  1)  sin"1"1  x      m  —  lj  smm~'ix  ^  ^ 

3  O 

/dx     _  sin  x  n  —  2Cdx  ^~<t 

cos"  re      (n  —  1)  cos""1  x      n  —  lj  cos"""2  x'  ^g 

u  S 

/tan"-1^        /" 
tan"x  .  dx  = r—  —    I  tan"~2  x  .  dx. 
n  —  L         J  _^_ 

ont""1  r          C  V> 

cot"  a;,  da;  =  ,        -   j  cotn~*x.dx.  : 

l~  J  EE 

-^ =  g& 

a+o  cos  x  ui  S 


v/ 
if  a2  > 


2  /A  /a  -  b           x\ 

tan"1!  V    — — r.tan-l, 

,2_ft2  \        a +6          2/' 


34  INTEGRAL  CALCULUS 


if  a2<&2. 


.  sin  a; . 


dx 


/- 

—  xm  cos  x  +m  I  re77*"1  cos  x  dx. 
I  xm  .  cos  x  .  dx  = 

xm  sin  x  —  m   I  re"1""1  cos  re  dx. 

/sin  re  ,  re3     ,     re5          re7     , 

—  dx=X-3l  +  5H-7]7  +  --- 

/sin  re  ^   _    —1    sin  x           1       fcos  re  a 
m°    ^*^ 1"       m. — \      ' T       I     m T 
xm            m  —  Lxm          m  —  \J     xm~ 

/cos  re,   _.  x2         x*         re8 

~x~°  '~2\2+4[4~"Q{6+' 

/cosx  ,          —  1      cos  re          1       /*sin  re  dx 
^dx=^l-^-^ij-^r- 

\  resin"1  re  .  dx  = 

^  [(2  re2  - 1)  sin"1  re  +rc  Vl-a;2]. 

/«"*>—.*- 

rcn  +  1sin-1a;          1       fxn  +  l  dx 
n  +  l  n  +  l  J  Vl-xz 

/n 
re- 

rcn+1  cos"1  re   ,      1       f ren+»  « 
n  +  l 


INTEGRAL  CALCULUS  35 

xn  tan"1  x  .  dx  — 


n+1           n  +  lj    1+z2 


dx 


*1 

3  O 

S3 


O  < 

z  cc 
<  LJ 

n 

2  u. 
O 


dx  2  .  xi . 

=  — sec"1 
/~n  _  /,2      an  a 


34  INTEGRAL  CALCULUS 


=.  log  - 


if  a2<62. 


ERRATA 


should  read 

J  -'•**  -  cos  .r  .  , 


INTEGRAL  CALCULUS  35 

I  xn  tan"1  x  .  dx  = 


n+1 


!«••  dr. 


*V""dfc-2-2--2  f  *•-! 

i/  «         aj 

fea*  ,         -1        ea*     ,      a       f  ea*    . 

—  dx=  -  -  .  —r—  -  -{  --  ;         —  —  -.dx. 
J  xn  n-l     xn~l      n-lj   xn~] 


/ax        /•      \  ^   —  ax\  a(cos  (nx^  +nsin(nrc)"l 
[  a?+n* 


/ 


dx 


THEORETICAL 
MECHANICS. 


NOTATION. 

A  =area. 

a  —  acceleration. 

an  =  normal  acceleration. 

at  —  tangential  acceleration. 

b   =  breadth. 

Cx  =  component   of  force    parallel   to   the 

X-axis. 
Cy= component  of    force  parallel   to   the 

Ct  =  component  of  force   parallel   to   the 

Z-axis. 
d  =  depth  or  distance.     Also  the  sign  of 

the  differential. 
F= force. 

Fn= normal  force  or  component  of  force. 
Ft  =  tangential    force    or    component    of 

force. 
/=  coefficient  of  friction.     Also  the  sign 

of  a  function  of  a  variable. 
g  =  acceleration    due    to    gravity  =  32.2. 
(The    exact    value    is    32.1808- 
0.0821    cos  2  L,    where    L    is    the 
latitude.) 
h  —  distance  from  center  of  moments  to 

line  of  force. 
7  =  moment  of  inertia. 
Iff  =  moment  of  inertia  referred  to  center 

of  gravity. 

Igx  —  moment  of  inertia  about  an  axis 
through  the  center  of  gravity  and 
parallel  to  the  X-axis. 


THEORETICAL  MECHANICS        37 

70  =  polar  moment  of  inertia  about   the 

pole  0. 

7a.= moment  of  inertia  about  the  Jf-axis. 
Iy  =  moment  of  inertia  about  the  Y-axis. 
/„  =  moment  of  inertia  about  the  Z-axis. 
J  =  product  of  inertia.      (Subscripts  are 

similar  to  those  for  /.) 
K=a  constant. 
L  =  power. 
M  =  moment  of  a  force. 

W 

m  =  mass  =  —  • 
Q 

N=SL  normal  force  or  component  of    a 

force. 

P  =  point  considered. 
R  =  resultant  of  a  system  of  forces. 
r  =  radius  of  gyration. 
s= space. 
T= tangential    force  or  component  of  a 

force. 
f  =  time. 
V— volume. 
v  =  velocity. 
VQ  =  initial  velocity. 
vt=  tangential  velocity. 
tf*= velocity  parallel  to  the  X-axis. 
vy= velocity  parallel  to  the  F-axis. 
W  =  weight. 
w  =  work. 

,  y,  z  =  rectangular  coordinates  of  a  point. 
p,  0  =  polar  coordinates  of  a  point. 

p  =  distance    from    pole    to     center    of 

gravity, 
a  =  angle. 
$==  angle  of  friction. 


38       THEORETICAL  MECHANICS 


STATICS. 
Equilibrium  of  Forces. 


Fig.  17. 

For  a  system  of  concurrent  forces  in  equi- 
librium in  one  plane: 


(Cz=F  cos  a,Cy=F  sin  a,  where  a  is  the 
angle  which  F  makes  with  X-X,) 


For  a  system 
of  non-concur- 
rent forces  in 
equilibrium  in 
one  plane  : 


Also,  i 


Fig.  18. 


If    three   forces    are    in    equilibrium    they 
must  be  concurrent  or  parallel. 


THEORETICAL  MECHANICS       39 

If  a  system  of  non-concurrent  forces  in 
space  is  in  equilibrium,  the  plane  systems 
formed  by  projecting  the  given  system  upon 
three  coordinate  planes  must  each  be  in 
equilibrium. 

A  couple  consists  of  two  equal  and  oppo- 
site parallel  forces  acting  on  a  rigid  body  at 
a  fixed  distance  apart. 

The  moment  of  a  couple  is  equal  to  the 
product  of  one  force  by  the  distance  between 
the  two  forces. 

Center  of  Pressure. 
FI,  F2,  F3,  etc.,  are  parallel. 


Fig.  19. 

If  F  is  the  force  exerted  by  a  variable 
pressure,  then 


xFdx 


Fdx 


40       THEORETICAL  MECHANICS 


Center  of  Gravity. 
For  an  area, 


x  dx  dy 


dxdy 


Fig.  21 


fx.dx 


THEORETICAL  MECHANICS       41 
If     x2-xi  =  1 

y= 


\  y  O2-Zi)  dy 
\  (x2-xi)  dy 

J  y  •  fy .  dy 


Fig.  22. 


For  a  homogeneous  mass, 

///*•*** 


x-  *xdm-. 
Zdm 


fSf 


dx  dy  dz 


-V => 


F 


Fig.  23. 
/-//•*** 


fff 
fff 


zdxdydz 


dxdydz 


42       THEORETICAL  MECHANICS 

Rectangular  Moment  of  Inertia. 
For  an  area, 

L=  C  fy^dxdy. 


Fig.  24. 


Fig.  25. 

=^dA 

=  j  7/2.  (xi-x 


THEORETICAL  MECHANICS       43 
If  Va-yi=fx, 


Fig.  26. 


=  (  x*  (yt—yi)  dx 

=  Cx2  .  f 


.  fx  .  dx. 


Fig.  27. 


Polar  Moment  of  Inertia. 
For  an  area, 


44        THEORETICAL  MECHANICS 
Since  p2=xz+y2, 


Fig.  28. 


For  a  mass, 


p*dxdydz 


•'//;• 

=kfff(x*+y*)dxdydz, 


Fig.  29. 

where  k  is  the  weight  per  cubic  unit  divided 
by  g. 

Product  of  Inertia. 
J  =  l 


.  y  .  dx  .  dy. 
Ji=Jo.f.+Akh, 


THEORETICAL  MECHANICS        45 

where  J\  is  the  value  of  J  referred  to  X  —  X 
and  Y  —  F,  Jo.%.  is  the  value  of  J  for  axes 
parallel  to  X  -  X  and  Y  —  Y  passing  through 
the  center  of  gravity,  and  h,  k  are  the  co- 


Fig.  30. 

ordinates  of  the  center  of  gravity  referred  to 
X-X  and  Y-Y. 

(See  "A  Complete  Analysis  of  General 
Flexure  in  a  Straight  Bar  of  Uniform  Cross- 
Section,"  by  L.  J.  Johnson,  Trans.  Am.  Soc. 
C.  E.t  Vol.  LVI,  1906.) 

Radius  of  Gyration. 


-  -. 

A  m 

Ellipsoid  of  Inertia. 

The  moments  of  inertia  about  all  axes 
through  any  given  point  of  any  rigid  body 
are  inversely  proportional  to  the  squares  of 
the  diameters  which  they  intercept  in  an 
imaginary  ellipsoid,  whose  center  is  the 
given  point,  and  whose  position  depends 
upon  the  distribution  of  the  mass  and  the 
location  of  the  given  point.  This  ellipsoid 
is  the  ellipsoid  of  inertia  for  the  body.  The 
axes  which  contain  the  principal  diameters 
of  the  ellipsoid  are  called  the  principal  axes 
of  the  body  for  the  given  point. 


46        THEORETICAL  MECHANICS 

DYNAMICS. 
Velocity  and  Acceleration. 

ds 
V  =  Jt' 

dr      cPs 


Uniformly  Accelerated  Motion. 
If  a  is  constant, 


2  a 


Falling  Bodies. 
For  a  body  falling  in  a  vacuum,  a  =  g,  hence 


Force  and  Acceleration. 

-  =  — 

~~  0  ' C 

Direct  Central  Impact. 
For  two  inelastic  bodies,  let 
mi  =  mass  of  first  body. 
m2  =  mass  of  second  body. 
Vi=  original  velocity  of  first  body. 
v2  =  original  velocity  of  second  body. 
v  =  common  velocity  after  impact. 


THEORETICAL  MECHANICS        47 

Then  v  =  miVl  +™2V2  . 

mi  +m,2 

For   two    elastic    bodies    having   velocities 
i  and  k2  after  impact, 


The  product  of  mass  by  its  velocity  is 
momentum. 

The  sum  of  the  momenta  before  and  after 
impact  is  constant. 

Virtual  Velocities. 
F  =  force. 

v  =  direction  of  motion  of  P. 
du  =  virtual  velocity  of  force. 

•fa  =  velocity  of  force. 

ds 

-^  =  velocity  of  P. 

F .  du  =  virtual  moment  of  force. 

The  virtual  moment  of  a  force  is  equal  to 
the  algebraic  sum  of  the  virtual  moments  of 
its  components. 

For  a  system  of  concurrent  forces  in 
equilibrium,  2  p  .  du  =  0, 

For  any  small  displacement  or  motion  of 
a  rigid  body  in  equilibrium  under  non-con- 
current forces  in  a  plane,  with  all  points  of 
the  body  moving  parallel  to  this  plane, 


Y 


Curvilinear  Motion  of  a  Point. 

ds 

v  Vt==dt' 

" /  clx  /rfs\2 

^_      *^» 

~ 1 V  y.7 


Y 


Fig.  3!. 


-(%f+m 


48        THEORETICAL  MECHANICS 


dv  _  d?s 
t      dt~  dP 

=  ax  cos  a  +Oy  sin  a. 
n  =  Oy  cos  a  —  a*  sin  a. 


where  r  is  the  radius  of  curvature. 

F=m  .  a,  .'. 


where  r  is  the  radius  of  curvature. 

Ft  =  m  .  ax  cos  a  +m  .  Oy  sin  a 


1V -/«•*• 


Projectiles. 
Neglecting  resistance  of  air, 

X  =  VQ  COS  OQ  .  t. 

y=vosm  OQ.  t-*%g&, 
-X, 


2  1'02  COS2  OQ  ' 


Horizontal  range, 

xr=  —  sin  2  oo, 

which  is  a  maximum  for  a0  =  45°. 
The  greatest  height  of  ascent, 


THEORETICAL  MECHANICS       49 


Translation  of  Rigid  Body. 
dFx—ax  .dm. 
R»—  I  «*•  dm. 


Fig.  33,  Fig.  34. 

The  resultant  force  must  act  in  a  line 
through  the  center  of  gravity  and  parallel  to 
the  direction  of  motion. 

Rotation  of  a  Rigid  Body. 

Let  O  be  the  axis  of  rotation. 
6  =  angular  space  passed  over  by  any  line 
from  O. 

a  =  angular   accelera-  "*xs 

tion.  / X  V 

a)  =  angular  velocity. 
Then 

dO 


= 
dt       dt*  ' 


Fig.  35, 


For  uniform  acceleration,  a.  =  k,  .'. 


2a 


.t. 


50       THEORETICAL  MECHANICS 
For  a  point  p  distant  from  O, 


p  .  ai. 
p  .  a. 


Fig.  36. 


For  a    mass    m    concentrated    p    distant 
from  O, 


Center  of  Percussion  or  Oscillation. 

If  an  unsupported  bar  upon  being  struck 
at  a  begins  to  rotate  about  6,  then  a  is  the 
center  of  percussion  for  6  as  a  center,  and  b 
is  the  center  of  instantaneous  rotation. 


Fh  = 


dF=a  .  p  .  dm. 

F=a  l  P.  dm 

=  a  .  p.m. 

^dHv  Fie-37- 


THEORETICAL  MECHANICS        51 


Pendulum. 

*  =  time  of  oscillation 
from  one  extreme  posi- 
tion to  the  other. 

r  =  radius  of  gyration. 

Then 


Work,  Energy,  and  Power. 

Work  is  equal  to  the  product  of  the  force 
by  the  distance  through  which  it  acts. 


Power  is  the  rate  of  doing  work. 


1  H.P.  =  33,000  ft.-lb.  per  min.  =  550  ft.-lb. 
per  sec. 

Energy  is  the  capacity  or  ability  to  do 
work. 

K.E.  =  Energy  of  a  moving  body. 


For  rotation, 


Fig.  39. 
Angle  of  friction, 


K.E.  =  £/.<o2. 

Friction. 

F= friction. 
N  =  normal  force. 
/= coefficient  of  fric- 
tion. 
F=f.N. 


pi 

^-tan-1:. 


52       THEORETICAL  MECHANICS 

Average    values    of    /   for    motion    are   as 
follows: 

Wood  on  wood       25-. 50 

Metal  on  wood 50-. 60 

Leather  on  metal 0 . 56 

Leather  on  metal,  lubricated   ...  0.15 

Metal  on  metal, — dry 0.15-.24 

Lubricated  surfaces: 

Ordinary 0.08 

Best 0.03-0.36 

For  values  of  /  for  rest  add  40  per  cent  to 
above  values. 


Fig.  40. 
dF=f.Nds 


,  where  0j  is  in  radians. 


MECHANICS   OF 
MATERIALS. 


w 
•8 

I 

i 


iO»OOC 

i-KM  COO 


o   . 

d^ 

03  GQ 


'•"A    I    X* 


t<  10  »O  iO  CO  00 


II 

H  S3 


IO^OI-KNT-IC* 

(N  CO  CO 


00  O  O  O  TH  rH  iH  u 


0  O  O 

-lTfiiO 


DOOO     •     •       O  00  00  W  OS  iO  W 

SllOCDO  iH  '-'         rHrH 


"££  -3  PL,  3  "3      U 

ftB^filPill 


53 


54      MECHANICS  OF  MATERIALS 


NOTATION. 

A=  area. 
6  =  breadth. 
d=  depth. 

Z£=modulus  of  elasticity. 
e  =  total  deformation. 
F=force  of  load. 
/  =  moment  of  inertia. 
/o  =  polar  moment  of  inertia. 
J  =  product  of  inertia. 
Z  =  length. 
M=  moment. 
R  =  resultant  of  forces. 
r  =  radius  of  gyration. 
£  =  unit  stress. 
s  =  section  modulus. 
V=  vertical  shear. 
JF=  total  weight. 
w  =  weight  per  lineal  unit, 
A  =  maximum  deflection. 
e=unit  deformation. 

Direct  Stress. 


I 

s 


Fl 
— —  . 

eA 


MECHANICS  OF  MATERIALS       55 


Fig.  43. 

Consider  a  section  a  —  a  perpendicular  to 
axis  of  a  bar,  and  take  axes  of  coordinates 
through  center  of  gravity. 

Let  xt  y  —  coordinates  of  any  point  of 
section. 

n  —  n  =  neutral  axis. 

v  =  distance  of  any  point  from  line  through 
center  of  gravity  and  parallel  to  neutral 
axis,  positive  toward  P. 

•yo  =  value  of  v  for  neutral  axis. 

F=  force  or  resultant  of  forces  acting  at  P. 

N  =  component  of  F  normal  to  section 
considered. 

£o  =  unit  stress  at  center  of  gravity. 


*  The  method  here  presented  is  taken 
from  a  paper  by  L.  J.  Johnson,  M.  Am.  Soc. 
C.  E.,  "An  Analysis  of  General  Flexure  in  a 
Straight  Bar  of  Uniform  Cross  Section," 
Trans.,  Am.  Soc.  C.  E.,  volume  LVI,  p.  169, 
1906. 


56      MECHANICS  OF  MATERIALS 

S  =  S0 *  .  v 

=  So (y  cos  a  —  x  .  sin  a) 

jy      N.  xp(y  —  xtan  a) 
A-  J  —  Iy  tan  a 

jy      N  .yp(y  —  x  tan  a) 

Ig  —  J  tan  a 

jY      N  .  pp(y—x  .  tan  a)  cos  0 
-4  J  —  Iy  .  tan  a 

N  .  pp(y  —  x  .  tan  a)  sin  0 


/»-,/.  tan  a 

AT      N(yPIy-xpJ)y+N(xjJae-yPJ)x 


lysin  6  —  J .  cos 0)  y+(Ix cos  0  — ./sin 


*)af| 


./V 

In  the  above  equations  -r-  is  the  portion  of 
.A 

S  which  is  direct  stress,  and  the  other  term 
is  the  portion  due  to  the  bending  moment, 
M  =  N .  pp.  If  s  represent  the  section 
modulus 

V(/ysin0  —  J  .  cos0)  y  +(/«  cos  Q  —  J  .  sin  0)  x) 
then 

N  .  M 


NOTE.  —  The  values  of  the  section  modu- 
lus  given   in   the   handbooks   are   computed 

from  the  formula  s  =  - ,  which  is  the  value  of 

y 


MECHANICS  OF  MATERIALS      57 

s  for  .7  =  0  and  for  P  located  on  Y-Y.     For 
angles  and  Z-bars  J  does  not  equal  zero. 
In  the  above  equations, 


tan  a  = 


/«-«/.  tan  0 
J  —  Iy  tan  0 

I x  cot  0-J 
J  cot  0-Iy 

Ix  cos  0  — «7  .  sin  0 
J  cos  0  —  lysin  0 


For  any  bar  having  a  section  which  is 
symmetrical  about  either  axis,  .7=0,  and 
the  values  of  S  become 


c 

= 


•  x\ 


If  for  a  symmetrical  section,  P  is  on  Y  —  Y, 
then  sin  0  =  1  and  cos  0  =  0,  or 


, 

o  —  ~:~  H 


.  PP  .  y 


Fig.  44. 

For  a  rectangular  section,  for  which  N  is 
applied  on  Y —  Y  and  p  distant  from  the 
axis  of  the  bar,  the  extreme  fiber  stresses  are 


£  =  ^1 


58       MECHANICS  OF  MATERIALS 


Equation  of  Neutral  Axis. 

The  equation  of   the  neutral  axis  for  an 
eccentric  load  is 

&P  *i*-yp.J\  iziy  -  J* 

V  ~  \Xp.J -yp.  Iy)X      A  (xp.J-yp.Iy-)  ' 


Kernel  or  Core-Section. 

The  kernel  of  a  section  (sometimes  called 
the  core-section)  is  the  area  within  which  Pt 
the  point  of  application  of  the  resultant  of  the 
forces,  must  fall  in  order  that  the  stress 
shall  be  of  the  same  sign  throughout  the 
section.  It  is  the  area  bounded  by  the 
locus  of  the  P's  corresponding  to  a  series  of 
neutral  axes  touching  the  periphery  of  the 
section  but  never  crossing  the  section.  For 
every  side  of  the  section  there  will  be  an 
apex  of  the  kernel.  If  xa,  ya  and  Xb,  j/j  are 
the  coordinates  of  a  and  b,  which  are  two 
consecutive  vertices  of  the  section,  then  the 
coordinates,  Xab,  2/o6,  of  the  vertex  of  the  kernel 
corresponding  to  the  side,  ab,  of  the  section 
will  be 


Xab 


(xa  -Xb)  J  —  (ya  —  yb)  Jy 

=  --  —  -  •  - 


yob  — 


A  (xayb-xbya) 
If  ab  is  parallel  to  X  —  X,  then 

J  I 

Xab=~T~17'     v<*=--A — — 

jA.  •  |/a  A.   ,  ya 

If  ab  is  parallel  to  Y  —  Y,  then, 

Iy  J 

A.Xa'        V<>b  A.Xa 


MECHANICS  OF  MATERIALS      59 

The  radii  vectores  of  the  kernel  are  lengths 
which  for  any  0  need  only  be  multiplied  by 
the  area  of  the  section  (A)  to  give  the  sec- 
tion modulus 


\(Iy  sin0-J. 

but  these  lengths  must  be  considered  posi- 
tive it  measured  on  the  opposite  side  of  O 
from  P. 


Section  Modulus  Polygons. 

In  the  equation  S  —  -r  -\  --  (see  Eccentric 
A        s 

Loads),  s  is  the  section  modulus.  The  sec- 
tion modulus  polygon  is  the  polygon  the 
lengths  of  whose  radii  vectores  are  the 
graphical  representations  of  the  values  of  s 
for  extreme  fibers  for  successive  values  of 
0  from  0  to  360  degrees.  The  section  modu- 
lus polygon  is  a  figure  whose  sides  are  parallel 
to  the  sides  of  the  kernel  of  the  given  section 
but  which  lie  on  opposite  sides  of  the  center 
of  gravity  from  the  sides  of  the  kernel. 
The  most  general  value  of  s  is 


_  __ 

(lysin  0  —  J  cos  0)y  +  (Iycos  Q  —  J  .  sin  0)  x 

For    any    section    which    is    symmetrical 
about  either  axis,  s  becomes 


Iy  sin  0  .  y  +IX  cos  0  .  x 

For  any  symmetrical  section  for  which  P 
lies  on  Y  —  Y.  0  =  90°,  hence 


60       MECHANICS  OF  MATERIALS 

If  for  any  symmetrical  section  P  lies  on 
X-X,e  =  0°,  hence 


There  will  be  one  vertex  of  the  s-polygon 
for  each  side  of  the  polygon  bounding  the 
section.  If  xa,  ya  and  #5,  yb,  are  the  coordi- 
nates of  a  and  6,  two  consecutive  vertices  of 
the  bounding  polygon  of  the  section,  then 
the  coordinates  of  the  vertex  of  the  s-polygon 
corresponding  to  the  side  ab  of  the  bounding 
polygon  will  be 


If  ab  is  parallel  to  X-X, 

J  I* 

Xab=—  ,  yab  =  —' 

ya  y<* 

If  ab  is  parallel  to  Y-Y, 

Jy  J 

Xab=—,  yab=  —  > 

Xa  Xa 

For  sections  symmetrical  about  either 
X-X,  or  Y-Y,  J  =  0,  and  the  values  of  — 

ya 

and  —     can    be    found    in    the    handbooks 

za 

issued  by  the  steel  companies,  under  the 
column  marked  "Section  Modulus."  The 
vertices  can  then  be  plotted  and  connected 
by  straight  lines  to  form  the  s-polygon. 
From  this  s-polygon  the  values  of  s  for  any 
value  of  9  can  be  obtained  graphically. 

The  most  advantageous  plane  of  loading  for 
any  section  will  be  that  having  the  greatest 
value  of  s. 


MECHANICS  OF  MATERIALS      61 


DIAGONAL  STRESSES 


Fig.  45. 

F  =  axial  load. 

A  =area  of  section  normal  to  axis  of  bar 
n  —  n  =  any  diagonal  section. 

0  =  angle  which  n  —  n  makes  with  axis. 
£  =  unit  axial  stress. 
$3  =  unit  shear  along  plane  normal  to  axis. 
$n=unit  tension  or  compression  normal 

to  section  n  —  n. 
$«n==unit  shear  along  section  n  —  n. 

For  combined  direct  stress  and  vertical  shear, 
Sn  =  ~  (1  -cos  2  0)  +S. .  sin  2  9. 

S«n  =  IT  •  sin  2  0  +S,  .  cos  2  0. 


The  maximum  or  minimum  value  of  Sn 

S 
occurs  when  cot  2  0  = ,  and  is 


max.  /S»=  —  ; 


The  maximum  value  of  S8n  occurs  when 
S 


tan 20= 


-,  and  is 


62       MECHANICS  OF  MATERIALS 

For  axial  load  only,  >S8=0,  hence 


n=      .  sin  2  0  = 


sin  2  *• 


The  maximum   value   of   Sn  occurs   when 
0  =  90°,  and  is  then  the  unit  axial  stress. 
The  maximum  value  of  Sen  occurs  when 

Q          W 

0  =  45°,  and  is  -  01-75—  i  • 
*j         Z  A 


THIN  PIPES,  CYLINDERS,  AND  SPHERES. 


S  =  unit  stress  in  metal. 
t  =  thickness  of  metal. 
d  =  diameter. 
;p  =  unit     pressure     of 

liquid  or  gas. 
0  =  angle    which    the 
direction    of    P 
makes  with 
X-X. 


Fig.  46. 


For  the  transverse  stress   across  a  longi- 
tudinal section  of  a  pipe  or  cylinder, 

.  cos  0  =  %  p.  d. 

P*d 
-' 


For  the  longitudinal  stress  across  a  trans- 
verse section  of  a  pipe,  or  for  the  stress  in  a 
thin  hollow  sphere, 


o 


nd.t 


4t 


which  is  one-half  of  the  unit  transverse  stress 
in  a  pipe  having  the  same  diameter  and 
thickness. 


MECHANICS  OF  MATERIALS       63 
RIVETED  JOINTS. 


Fig.  47. 

a = distance  center  to  center  of  two  con- 
secutive rivets  in  one  row. 
d= diameter  of  rivet  or  rivet  hole. 
F— stress  in  unriveted  plate  in  length  a. 

t  —  thickness  of  plate. 
S$=unit  tensile  stress. 
$c=unit  compressive  or  bearing  stress. 
£«  =  unit  shearing  stress. 
et= efficiency  of  joint  for  tension. 
ea= efficiency  of  joint  for  compression. 
et  =  efficiency  of  joint  for  shear. 
m  =  number  of  shearing  sections  of  rivets 
in  distance  a.     (Notice  that  for  butt 
joints  each  rivet  has  two  shearing 
areas.) 

n— number  of  bearing  areas  of  rivets  in 
distance  a. 

F  =  t  (a  —  d)  Sf = m  .  —  xd?  .  St~n  •  t  •  d  •  Se. 


m.  x.  d?Sg 

4 .  atSt 
n .  dSe 


64       MECHANICS  OF  MATERIALS 

For  maximum,  efficiency,  make  et  =  et 
for  which 

4  .  n  .  Se .  t 
m.n  .  SB 

and      a  —  —  ~[l  +n  -3-1 1 . 


For  single  riveted  lap  joints  the  maximum 
efficiency  is  approximately  55  per  cent,  for 
double  riveted  lap  joints  approximately  70 
per  cent,  for  triple  riveted  lap  joints  approx- 
imately 75  per  cent,  and  for  triple  and 
double  riveted  butt  joints  approximately  80 
per  cent. 

BEAMS. 

Vertical  Shear.  The  vertical  shear  at  any 
given  section  of  a  horizontal  beam  is  the 
sum  of  the  vertical  components  of  all  of  the 
stresses  at  that  section.  The  vertical  shear 
is  equal  to  the  sum  of  all  the  reactions  of 
the  supports  upon  the  left  of  the  given  sec- 
tion minus  the  sum  of  all  of  the  vertical  loads 
on  the  left  of  the  section. 

For  any  beam  the  vertical  shear  upon  the 
right  side  of  the  left  support  of  any  span  is 


where 

MI  =  the  moment  at  the  left  support, 
M2  =  the  moment  at  the  right  support, 
10  =  the  uniform  load  per  lineal  unit, 
jP=any  concentrated  load, 
a  =  the  distance  from  the  left  support  to  F, 
Z  =  the  length  of  span. 


MECHANICS  OF  MATERIALS       65 

Shearing  Stresses.     If  F  =  vertical  shear  at 
any  section, 

- 


where  S9  is  the  average  unit  shear. 

The  actual  unit  vertical  shear  at  any 
point  is  equal  to  the  unit  horizontal  shear  at 
that  point,  and  may  be  determined  by  the 
following  equation: 


where  6  is  the  breadth  of  the  section  at  the 
given  point,  y  is  the  distance  of  the  point 
considered  from  the  neutral  axis,  and  c  is 
the  distance  from  the  neutral  axis  to  the 
extreme  fiber  on  the  same  side  as  the  point 
considered. 

The  maximum  value  of  S,  occurs  at  the 
neutral  axis,  and  is 


V     C°  V 

max.S.=  j- -^   I    y. 
*     **  o 


l.b 


where  AI  is  the  area  of  the  portion  of  the 
section  on  one  side  of  the  neutral  axis,  and 
yi  is  the  distance  from  the  neutral  axis  to 
the  center  of  gravity  of  the  portion  of  the 
section  on  one  side  of  the  neutral  axis. 

For  a  rectangular  section,  the  maximum 
unit  shear  is  |  of  the  mean  unit  shear. 

For  Diagonal  Shear,  see  Diagonal  Stresses, 
page  61. 

Bending  Moment.  The  bending  moment 
at  any  point  for  any  beam  is 

M  =  Mj  +  Vlx-$wx*-'2F  (x-a), 

*  See  "Merriman's  Mechanics  of  Materi- 
als," page  269. 


66       MECHANICS  OF  MATERIALS 

where 

M  =  bending  moment  at  section  considered, 
MI  =  bending  moment  at  the  left  support, 
FI  =  vertical  shear  upon  the  right  side  of 

the  left  support, 
w  =  uniform    load    including     weight    of 

beam,  per  lineal  unit, 
F=any  concentrated  load  upon  the  left 

of  the  section  considered, 
x  =  distance  from  the  left  support  to  the 

section  considered, 
o  =  distance  from  left  support  to  F. 

For  any  beam  of  one  span  Vi  is  equal  to 
the  reaction  at  the  left  support. 

The  maximum  values  of  M  occur  at  those 

sections  for  which— 7—  =0,  that  is,  where    the 

dx 

shear  passes  through  zero. 

The  values  of  M  for  special  cases  are  given 
in  Table  of  Beams,  page  68. 

Theorem  of  Three  Moments.  For  any  two 
consecutive  spans  of  a  continuous  beam,  let 

MI  =  moment  at  the  left  support, 
M2= moment  at  the  middle  support, 
M 3  =  moment  at  the  right  support, 
Zi=length  of  the  first  span, 
h  =  length  of  the  second  span, 
1  =  length  of  span  for  equal  spans, 
wi  =  uniform  load  per  lineal  unit  on  first 

span, 
w 2  =  uniform  load  per  lineal  unit  on  second 

span, 
FI  =  any    concentrated    load    on    the   first 

span, 
F2  =  any  concentrated  load  on  the  second 

span, 

«i=  distance  from  first  support  to  FI, 
d2  =  distance  from  middle  support  to  F%. 


MECHANICS  OF  MATERIALS       67 

Then,  for  uniform  loads  only, 

ih  +2  M2  (h  +  12)  +M312  =  -$w1ll3-iw2l2*. 

For  equal  spans  with  equal  uniform  loads, 


For  concentrated  loads  only, 

Mill  +2M2  (h  +fe)  +M312 


Flexural  Stresses.  The  tensile  and  com- 
pressive  stresses  in  a  beam,  produced  by 
bending,  are  the  same  as  the  stresses  upon  a 
section  having  an  eccentric  load,  due  to  the 
moment  of  that  load.  Therefore,  for  pure 
flexure  the  tensile  and  compressive  stresses 
for  the  extreme  fibers  of  any  section  can  be 

determined  by  placing  —  =0  in  the  formula 

for  S  given   under  Eccentric  Loads,    which 
gives 


where  s  is  the  section  modulus,  the  values  for 
which  are  given  under  Section  Modulus 
Polygons. 

For  combined  flexure  and  direct  stress,  the 
tensile  and  compressive  stresses  are  given  by 
the  formulae  for  Eccentric  Loads. 

Elastic  Curves.  The  curve  \vhich  is  as- 
sumed by  the  neutral  surface  of  a  beam 
under  load  is  called  the  elastic  curve. 

The  radius  of  curvature  of  the  elastic  curve 
is 

ff/=      dP      ^dtf 
'     M      dx.d*y      d*y' 


68       MECHANICS   OF  MATERIALS 

from  which  the  equation  of  the  elastic  curve 
can  be  obtained,  for  any  particular  case,  by 

placing    M  equal  to  El  •— ,  and  by  making 

two   integrations   to   obtain   an    equation   in 
terms  of  x  and  y. 

The  deflection  of  a  beam  at  any  given 
point  is  obtained  by  substituting  the  par- 
ticular value  of  x  in  the  equation  of  the 
elastic  curve  and  solving  for  y.  The  maxi- 
mum deflection  occurs  at  the  section  for 
which  #y 

dx~ 

(For  particular  cases,  see  Table  of  Beams.) 


TABLE  OF  BEAMS. 

NOTE  .  —  The  equations  for  elastic  curves 
and  the  values  of  A  apply  only  to  beams  of 
uniform  section. 

Beams  Supported  at  Both  Ends  and  Uniformly 
Loaded. 


Moment 


Fig.  48. 


MECHANICS  OF  MATERIALS      69 


=Rix-     wx* 

«     wlx  -     wx* 


A  when  x=  -  ,  or 


A==oo3  "Fr  ^QQA" 
384  til      o84 


Beam  Supported  at  Both  Ends  and  Loaded 
with  a  Concentrated  Load  at  Center  of  Span. 

f 


Ri 


Moment 


Fig.  49. 


70      MECHANICS  OF  MATERIALS 

V=Rlt  or  V  =  R2. 
M  =  Rtx,  on  the  left  of  F, 

=  Rix-Fx--ton  the  right  of  F. 


EI  l=  Fx>  on  the  Ieft  of  F- 

48  EIy  =  F  (4  *'-3  Pa;),  on  the  left  of  F. 


A=__ 

'     48  J^/  ' 

(For  both  uniform  and  concentrated  loads, 
combine  the  results  for  each.) 

Beam  Supported  at  Both  Ends  and  Loaded  with 
a  Concentrated  Load  Distant  a  from  the  Left 
Support. 


MECHANICS  OF  MATERIALS      71 

V=Ri,  on  the  left  of  F, 
=  #2,  on  the  right  of  F. 

M=Rix,  on  the  left  of  F, 

*=Rix-F  (x-a),  on  the  right  of  F. 


EI~~  =  Rlx,  on  the  left  of  F, 

=  Ri.x-F  (x-a),  on  the  right  of  F. 
Ely  =  -  Rix*  +  cix  4-  c2,  on  the  left  of  F, 


Fax* 
on  the  right  of  F. 


The  maximum  deflection  (A)  occurs  at  the 
section  for  which 


and  is 


Beam  Supported  at  Both  Ends  and  Loaded 
with  Several  Concentrated  Loads. 


72       MECHANICS  OF  MATERIALS 

The  maximum  moment  (Mm)  occurs  at 
the  section  for  which  R\  —  2  ^=0,  that  is, 
where  the  vertical  shear  is  zero. 

For  a  system  of  movable  loads  the  maxi- 
mum moment  will  occur  under  one  of  the 
loads,  the  loads  being  in  such  a  position 

,F1      !F*         !F3 


•  • 


Fig.  51. 


that  the  center  of  the  span  is  midway  be- 
tween the  center  of  gravity  of  all  the  loads 
and  the  section  at  which  the  maximum 
moment  occurs. 

The  maximum  deflection  of  a  beam  loaded 
with  several  loads  is  the  sum  of  the  deflec- 
tions produced  by  each  load  at  the  section 
at  which  the  maximum  deflection  for  the 
entire  system  of  loads  occurs.  The  deflec- 
tions produced  by  each  load  can  be  obtained 
by  means  of  the  equation  of  the  elastic  curve 
for  a  single  load. 


MECHANICS  OF  MATERIALS       73 

Cantilever  Beam  with  Uniform  Load. 
Rl  =  wl=W 
#2=0. 


Moment 

Fig.  52. 

Of  if  £  is  taken  from  the  free  end, 

1 

~2WX' 


El  0  =  i  wP-'  wlx  +  i  wx*. 


4  -  4 


I  wl*  _l  Wl3 
8  El  "8  El  ' 


74      MECHANICS    OF  MATERIALS 

Cantilever   Beam  with    Concentrated  Load  at 

the  Free  End. 
Ri  =  F. 

#2=0. 


3  El' 


Beam  Fixed  at  Both  Ends  and  Uniformly 
Loaded. 


M  =  —  — 


-  wlx  —  2 


MECHANICS  OF  MATERIALS       75 


By  placing  ~  —  0  when  x =0  and  when  x*=lt 


w  (  - 

JLE 

384  £;/ 


-re4). 


Beam  Fixed  at  Both  Ends  and  Loaded  at 
the  Center  of  the  Span  with  a  Concen- 
trated Load. 


V  =  Ri,  on  the  left  of  F, 
=*  R2,  on  the  right  of  F. 


76       MECHANICS  OF  MATERIALS 


M  =  -    FI  +    FX,  on  the  left  of  F, 

a  & 


on  the  right  of  F. 


'  Moment 


Mo 


Fig.  55. 
+  |F*,  on  the  left  of  F. 


on  the  right  of  F. 

By  placing  -     =  0  when  a:  =  0  and  when  x  =     . 


MECHANICS  OF  MATERIALS       77 


48  Ely 


,  on  the  left  of  F. 


1    FP 

''  192  El  ' 


Beam  Fixed  at  Both  Ends  and  Loaded  with 
a  Concentrated  Load  Distant  a  from  the 
Left  Support. 


Fig.  56. 


V  —  RI,  on  the  left  of  F, 
=  #2,  on  the  right  of  F. 

M  =  Mi  +Rix,  on  the  left  of  F, 

=  MX  +  Rix  -  F  (x  -  a),  on  the  right  of  F. 


78       MECHANICS  OF  MATERIALS 


7       =  Afi  +  Rix,  on  the  left  of  F. 

6  Ely  =  3  M  !X2  +#i:r3,  on  the  left  of  F. 

The    maximum    deflection    (A)    occurs  at 
2al 


the  section  for  which  x  - 


l+2a 


El  (I  +2  a)2      3  El  (I  +2  a)3 

Continuous  Beam  with  Uniform  Loads. 
u?j=load  per  lineal  unit  on  Zj. 
w2= load  per  lineal  unit  on  12,  etc. 
TFi  =  total  load  on  Zi. 
TF2  =  total  load  on  I2.t  etc. 


Rn=Vn. 

For  a   continuous   beam   supported  at  the 
ends, 

O+2M2  (h 


MECHANICS   OF  MATERIALS      79 
M212  +2  M3  (la 

=  —  -  w2l2z  —  j  wda-,  etc. 


=  —  -  wn-  2ln-  2*  —  r  w»-i/n-i2. 

From   the   above   simultaneous    equations 
*,  .  .     Mn-\  can  be  determined. 


1*2  tR3  tR3 

*i« — zi — -F — ti — r — ^H 


Moment     X^X 


V 


Fig.  57. 


For  equal  spans  with  equal  uniform  load 
over  the  entire  beam,  the  ends  of  the  beam 
resting  upon  supports,  the  moment  at  any 
support  is  KwP  or  KWl,  and  the  vertical 
shear  is  Nwl  or  NW,  where  K  and  N  have 
the  values  given  in  the  following  table: 


80      MECHANICS   OF  MATERIALS 


2 


o     o     o     o     o 


-S   HS 


a 

<O 

J4 

a 


«« 


MECHANICS   OF  MATERIALS      81 

For  a  continuous  beam  with  fixed  ends  con- 
sider an  imaginary  span  to  be  added  at  each 
end  of  the  beam,  with  the  free  ends  resting 
upon  supports.  Then  write  the  equation  of 
three  moments  for  each  two  consecutive 
spans,  making  1  =  0  for  the  first  and  last 
spans,  and  compute  the  moments  at  the 
supports  as  shown  above. 

Continuous  Beams  with  Concentrated  Loads. 

Determine  the  moments  at  the  supports  in 
a  similar  manner  to  that  employed  for  con- 
tinuous beams  with  uniform  load,  employing 
the  equation  of  three  moments  for  concen- 
trated loads. 

STRUTS  AND  COLUMNS. 

Euler's  Formula, 


Fig.  58. 


.-Vf  .sin-CO-cr 
y=a  .  sin  (*V  -§)  . 


82      MECHANICS  OF  MATERIALS 


7  J  /     Tf 

Since  y  =  a  when  x  —  -  ,    -  v   TTT  must  equal 
2      2        til 


—  =  7tzE  \j\  ,  for  round  ends. 

For  one  end  round  and  the  other  end  fixed, 

4. 
replace  I  by  -  1  and  TT  by  2  n,  which  gives 

o 


o 

For  6oiA  ends  /ixed,  replace  Z  by  -  I  and  ^ 

by  3  it,  in  the  formula  for  round  ends,  which 
gives 


Rankine's     Formula.      (Sometimes     called 
Gordon's  Formula.) 


Fig.  59. 

From  the  formula  for  eccentric  loads  for 
a  symmetrical  section  (page  57),  the  maxi- 
mum stress  will  be 


MECHANICS   OF   MATERIALS      83 

where  y  is   the   distance  from   the   neutral 
axis  to  the  extreme  fiber. 

But,  I^Ar2,  M  =  Fa    and  a  =  K-  ,  where 

y 

K  is  some  constant  depending  upon  charac- 
ter and  condition  of  the  column.     Hence 


Cambria    handbook    gives    the    following 
values  of  K  for  steel  columns: 
1 


36,000 


for  both  ends  fixed, 


24^00  f°r  one  end  fixed, 

,  for  pin  connected  ends. 


18,000 

The   above   values   are   to   be   used    with 
following  values  of  S  for  ultimate  strength: 
5  =  50,000  for  medium  steel. 
S  =  45,000  for  soft  steel. 

Hitter's  Formula.     Hitter's  formula  is  the 

same  as  Rankine's  formula  except  that  the 
o 

expression  —  is  used  for  K  ,  in  which  Se  is 
nE 

the  elastic  limit  of  the  material,   and  n  is 

g 

equal  to  n2  for  round  ends,  —  n?  for  one  end 

round  and  one  end  fixed,  and  4  rc2  for  both 
ends  fixed. 

The  Straight  Line  Formula.     The  straight 
line  formula  is 


where  C  is  a  constant  depending  upon  the 
character  and  condition  of  the  column. 


84      MECHANICS   OF   MATERIALS 

Merriman  gives  the  value  of  C  in  the 
above  equation  to  be 

C  =  —  Sf  (   ^J    \  t 

which  is  obtained  by  making  the  straight  line 
a  tangent  to  the  curve  for  Euler's  formula 

passing  through  the  point  S  for  -  =  0. 

The  following  values  of  S  and  C  for  allow- 
able stresses  are  given  in  Cooper's  Specifica- 
tions for  Railroad  Bridges,  1906. 

5  =  10,000,     C  =  45, 

for  live  load  on  chords, 
5  =  20,000,     C  =  90, 

for  dead  load  on  chords, 
5=   8,500,     C  =  45, 

for  live  load  on  posts  of  through 

bridges, 
5  =  17,000,     (7=90, 

for  dead  load  on  posts  of  through 

bridges, 
5=  9,000,     C7=40, 

for    live    load    on    posts    of    deck 

bridges, 
5  =  18,000,     (7=80, 

for   dead    load   on   posts   of   deck 

bridges, 
5=13,000,     (7  =  60, 

for  wind  load  on  lateral  struts. 

Engineering  News  Formula.  The  Engin- 
eering News,  Vol.  57,  No.  1,  Jan.  3,  1907, 
gives  the  following  formula: 


which  is  the  same  as  Rankine's  formula  given 
on  page  87,   allowing  the  eccentricity  a  to 


MECHANICS  OF   MATERIALS      85 

remain  in  the  formula  instead  of  substituting 
K-.  The  value  of  a  to  be  used  may  be 

considered  to  represent  the  eccentricity  due 
to  imperfection  in  manufacture  (since  it  is 
impossible  to  obtain  the  ideal  straight 
column),  plus  the  additional  eccentricity 
due  to  the  failure  to  obtain  an  axial  load. 
The  proper  value  of  a  to  obtain  correctly 
proportioned  columns  might  be  determined 
empirically  by  experiment,  or  it  may  be 
determined  by  comparison  with  column 
formulae  in  use  which  have  been  found  to 
give  correct  results. 

For  any  formula  of  the  Rankine  type, 


y 

In  the  article  above  mentioned  the  values 
of  a  for  a  number  of  formulae  in  use  are 
thus  computed,  the  mean  values  being  as 
follows: 

a  =  0.000051-,  for  steel, 

y 

a  =  0.000177-  ,  for  cast  iron, 

72 

a  =  0.000164-  ,  for  timber. 
For  any  formula  of  the  straight  line  type 
CAlr 

~ 


In  the  article  above  mentioned  the  values 
of  a  for  a  number  of  formulae  of  the  straight 

line  type  have  been  computed,  using-  =0.8, 
the  mean  values  being  as  follows: 

a  =0.0053  I,  for  steel, 

a  =  0.0015  I,  for  cast  iron. 

a  =0.0044  I,  for  timber. 


86      MECHANICS   OF   MATERIALS 

Eccentrically  Loaded  Columns.  To  the 
quantity  a  in  the  Engineering  News  formula 
add  the  eccentricity  of  the  load  at  the  end 
of  the  column,  that  is 


'-ll1 


(e+an/l 


where  e  =  eccentricity  of  load  at  the  end  of 
the  column. 

To  determine  the  maximum  stress  of  an 
eccentrically  loaded  column  by  Rankine's 
formula  replace  a  in  the  above  formula  by 

p 
its  equivalent  K  - ,  which  gives 


TORSION. 

Circular  Sections. 
Twisting  moment,  M  =  Fa. 

Circular  Sections 


Fig.  60.  Fig.  61. 

Resisting   moment,    Mr—  I  —SdA,  where 
S  is  the  shearing  stress  at  the  extreme  fiber. 
M =Mr,  or 

M—  ^ 
M~  R  ' 

where  70  is  the  polar  moment  of  inertia. 


MECHANICS  OF  MATERIALS      87 


For  a  solid  round  shaft    7?  =  -   nR3,  hence 

K       2, 


Non-Circular  Sections.  (Taken  from  Mer- 
riman's  "Mechanics  of  Materials.")  For 
non-circular  sections  the  above  formulae  are 
only  approximate. 

For  an  elliptical  section  whose  major  axis 
is  m  and  whose  minor  axis  is  n  the  maximum 
stress  is 

WFa 


For  a  rectangular  section  whose  long  side 
is  m  and  whose  short  side  is  n,  the  maximum 
stress  is 

,     9  Fa 


Transmission  of  Power.     The  horse-power 
which  is  transmitted  by  a  shaft  is 


" 


i 
550X12 

where  a  =  moment  arm  in  inches, 

<«>==  number  of  revolutions  per  sec. 

But,    Fa   =^,  hence 


INDEX 


PAGE 

Acceleration 46 

Analytic  Geometry 11 

Arithmetical  Progression 2 

Beams 64 

Continuous  Beams 78 

Coefficients  for  Continuous  Beams     .    .  80 

Table  of  Beams 68 

Theorem  of  Three  Moments 66 

Bending  Moment 65 

Belt,  Friction  of 52 

Binomial  Theorem 3 

Calculus 20 

Catenary,  The 18 

Center  of  Gravity .    ,  40 

Center  of  Pressure 39 

Circle,  The 14 

Columns 81 

Combinations  and  Permutations  ....  3 

Cones,  Equation  of 18 

Conic  Sections,  General  Equation  of    .    .  19 

Core  Sections 58 

Couple,  Definition  of 39 

Curves,  Elastic ,    .    .            .  67 

Cycloid,  The 17 

Cylinders,  Stresses  in       62 

Deflection  of  a  Beam 68 

Determinants ,    .    .    .    .  4 

Differential  Calculus 20 

Differentiation 21 

Dynamics 46 

Elastic  Curves      67 

Ellipse,  The 15 

Ellipsoid  of  Inertia 45 

Energy  .    , 51 


90  INDEX 

PAGI 

Equilibrium      ,  o£ 

Exponents ] 

Falling  Bodies      tj 

Force ^ 

Friction r,] 

Friction  of  Belt 55 

Geometric  Progression 5 

Harmonic  Progression 2 

Hyperbola,  The ie 

Hyperboloids ig 

Impact 4(3 

Integral  Calculus 23 

Kernel  of  a  Section 58 

Logarithms 1 

Materials,  Strength  of 53 

Maximum  Value  of  a  Function 22 

Mechanics,  Theoretical 3i} 

Minimum  Value  of  a  Function      ....  22 

Moment,  Bending 65 

Moment  of  Inertia  - 42 

Moment,     Maximum    for    Concentrated 

Loads 70 

Motion  of  a  Point,  Curvilinear 47 

Neutral  Axis,  Equation  of 58 

Parabola,  The 15 

Parabola,  The  Cubic '.    .  18 

Paraboloids 19 

Pendulum 51 

Permutations  and  Combinations  ....  3 

Pipes,  Stresses  in 62 

Plane  Triangles 8 

Power 51 

Power,  Transmission  of 87 

Product  of  Inertia 44 

Progression 2 

Projectiles 48 

Proportion 2 

Quadratic  Equations 2 

Radius  of  Curvature 22 

Radius  of  Curvature  of  Beams 67 


INDEX  91 

PAGE 

Radius  of  Gyration 45 

Riveted  Joints 63 

Rotation 49 

Section  Modulus 59 

Series 3 

Shear,  Diagonal 65 

Shear,  Vertical 64 

Shearing  Stresses 65 

Sphere,  The 18 

Spherical  Triangles 9 

Spheroids 18 

Spirals 17 

Statics 38 

Straight  Line,  The 13 

Stresses  — 

Combined  Stresses 67 

Diagonal  Stresses 61 

Direct  Stresses 54 

Eccentric  Stresses 55 

Flexural  Stresses 67 

Stresses     in     Pipes,     Cylinders,     and 

Spheres 62 

Struts 81 

Taylor's  Theorem 22 

Theorem  of  Three  Moments 66 

Torsion 86 

Transformation  of  Coordinates      ....  11 

Transmission  of  Power 87 

Triangles,  Solution  of 8 

Trigonometry 5 

Velocity  and  Acceleration 46 

Velocities,  Virtual 47 

»Vork  and  Energy 51 


^3»*€ijR^ 

OF  THE 

[UNIVr 

Oi 


c 
w 


a 

o 

i 


> 
r 
53 

I 


